ISL 2005 G6

Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$ . Let $\gamma$ be the incircle of triangle $ABC$ . The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$ . Let the lines passing through $K$ and $L$ , parallel to $BC$ , intersect the incircle $\gamma$ again in two points $X$ and $Y$ . Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$ . Prove that $BP = CQ$ .